If the computer were really smart, it would say, "Interesting. Yes, I can do that, but it will take some time. Seven and a half million years." Then it will relax while appearing to give the problem deep thought.

If the computer were really smart, it would say, "Interesting. Yes, I can do that, but it will take some time. Seven and a half million years." Then it will relax while appearing to give the problem deep thought.

You're absolutely right: pi is irrational, and as such, there won't be any repeats. However, that doesn't mean there isn't a pattern. For example, 0.12112111211112111112... is irrational, but there's a clear pattern that you could extend to an infinite number of digits. Does such a pattern exist once you get to a certain number of digits in pi? We don't know.

Firstly, just because something isn't repeating doesn't mean there isn't a pattern.
1,2,4,8 isn't repeating, but the pattern is there. (Each number doubles the previous)
1,1.5,2.25,3.375 also doesn't repeat but there is a pattern.(Each number is the previous number plus half the previous number)

Knowing (thinking) that something doesn't repeat and PROVING that it doesn't repeat are two ENTIRELY different things. I am guessing your maths/science education either stoppe

We know without a doubt that it never repeats - if it did it would be a rational number, it has been proven to be an irrational number, moreso it is transcendental. We also know the exact pattern, take the taylor series of sin about pi/4, you get an elegant and simple series solution for pi.

That is not the point. The point is and exercise in computing, everything we do in computing involves rational numbers only (floats) and there is substantial error involved with this. It is computationally difficult to deal with large numbers, hence any method to do this more effectively is a gain for science.

If there are interesting patterns in Pi, it'll be discovered through analytical research, not calculating digits out to some indeterminate end. I mean honestly, do they think the 2.5 trillion and one digit is going to hold the secret to one of the simplest shapes in mathematics?

No, but you may be surprised at how much mathematics is done computationally today. Many number theorists, for instance, spend an inordinate amount of time writing computer programs with the general intention of finding the answer first and determining the reason (i.e. proving it) second.

Knowing (thinking) that something doesn't repeat and PROVING that it doesn't repeat are two ENTIRELY different things. I am guessing your maths/science education either stopped very early or you didn't do too well in either.

I think it's funny that you are insulting someone's math education immediately after you imply that no proof exists showing pi not to repeat.

"Knowing (thinking) that something doesn't repeat and PROVING that it doesn't repeat are two ENTIRELY different things. I am guessing your maths/science education either stopped very early or you didn't do too well in either."

I wish I'd noticed this earlier, so I could belittle you proper. I'll leave it to you and your superior science/math brain to figure out why I find this amusing.

While I think that the computing horsepower was misdirected (covered elsewhere), and the last trillion digits could have waited, this post is mostly here for me to be arrogantly dismissive and make dick / vagina jokes.

Except that the ability to run 100m really fast is an enhancement we will eventually want to give to our soldiers before we run them off to Planet P to capture the brain. (Side Note: I want to see how fast we can get people to run *with* performance enhancers made legal)

On the other hand the ability to calculate pi just proves that we have a computer capable of making a vast number of calculations. I could well run the same numbers through my GPU - it just might take a little longer - to prove the same thi

Otherwise it would mean other non-predictable numbers could actually be predictable, potentially make breaking cryptography easier (much like finding out that a prime really isnt), would generally disrupt a bunch of mathematical theorems probably pissing off a whole sect of mathematicians, and turn a lot of things we think we know upside down.

Cryptography has nothing to do with a prime "not being a prime". It's to do with quick factorization of primes.

Besides, I don't see why pi having any sort of repeating pattern would disrupt any theorems. I honestly can't think of any theorem that requires such a thing. Irrational and transcendental yes, but no repeating decimal pattern?

Your post is not informative. Please reference elliptic curve cryptography [wikipedia.org] for why research in this field might actually yield valuable insights in the field of crytography. If you can't grasp it after a cursory overview of the topic, you probably shouldn't have replied to the GP, even given the fact that s/he was obviously misguided on the whole prime-or-not concept.

Wrong. At multiple levels. First of all, no form of cryptography relies on the digits of Pi not having a "pattern"(whatever that means). There is cryptography that relies on the conjecture (note, not theorem, but conjecture) that factoring numbers into primes is difficult. More specially, it is conjectured that factorization cannot be done in polynomial time. However, there's nothing at all connected to the digits of Pi, nor is there is anything connected numbers that one might think are prime that aren't.

The article isn't really that informative. It takes things too literally, using the known size of the universe to determine the largest possible physical circle and the smallest possible length (planck length) to determine the maximum precision and he comes up with 50 digits. But it wouldn't be too hard to come up with an application that uses more than 50 digits of pi. A new encryption algorithm could use sequences in pi, but this has nothing to do with physical circles. Math is abstraction, and there are fields in math that are so abstract that you can't even correlate them with a physical measure. It's very silly to say that knowing pi to more that 50 digits is useless.

Since Pi is irrational, does that mean that a "perfect" circle cannot actually exist? If you don't understand my question, think about it like this. Let's say I want to construct a circle of radius R. To create a "perfect" circle, it seems like I would need a length of material to build the circle out of that was exactly 2*Pi*R, but since Pi is irrational, it seems that you could never actually get any length which is an exact multiple of Pi? If Pi really expands out infinitely, even a circle with a radius the size of a galaxy, or a cluster of galaxies, could never be *exactly* the right length?

Question for the mathematicians... Can it really be proven that Pi is irrational or did it just get that reputation since it is a number that has no known end? I understand that from the laws and proofs of maths certain numbers can't exist as rational numbers (the sqr root of a negative) but Pi, in my limited knowledge of math, doesn't seem to fit into that. Is there an easy way to determine if a number is irrational?

I believe you are confusing rational numbers and real numbers. rational numbers are those that can be expressed as p/q where p and q are prime integers.
The existence of real numbers that are not rational follows from cantor's diagonal argument : http://en.wikipedia.org/wiki/Cantor's_diagonal_argument [wikipedia.org]

Proofs of the irrationality of pi can be found on wikipedia : proof [wikipedia.org]

If you don't understand my question, think about it like this. Let's say I want to construct a circle of radius R. To create a "perfect" circle, it seems like I would need a length of material to build the circle out of that was exactly 2*Pi*R, but since Pi is irrational, it seems that you could never actually get any length which is an exact multiple of Pi?

In an ideal world? Just take a unit of material and roll it into a circle. You'll never be able to measure the radius exactly, but you'll have your

To travel from one point to another, an object must pass through all the points in between. There are an infinite number of points "in between," thus to move at all, an object must travel through an infinite number of points in a finite time. Clearly this definition of reality is flawed: stop using it.

To travel from one point to another, an object must pass through all the points in between. There are an infinite number of points "in between," thus to move at all, an object must travel through an infinite number of points in a finite time. Clearly this definition of reality is flawed: stop using it.

Not necessarily. We can't really know about anything smaller than the Planck length, so in practical terms your paradox probably fails. The universe may be discrete on those scales.

Mod parent up - AC or not... I had to scroll a LONG way before seeing this argument and was going to post it myself if no-one else had. There's a lot of "weird" points about the universe that just don't seem to make sense. Posts such as the GP saying, "Clearly this definition of reality is flawed: stop using it." (with regard to travelling through an infinite number of points in a finite time) are all well and good, but don't go anywhere towards explaining WHY this definition is flawed. By defining the universe as discrete rather than continuous, it is no longer flawed, as with many other oddities and apparent paradoxes.

This would also potentially have an interesting effect on Pi in that if the number itself is truly irrational, then it's also wrong for every case we're using it - we actually should HAVE TO round it off somewhere to be correct when using it in models of the physical universe.

You're assuming that the circumference of a circle will always have an irrational length. Not so. There's no reason you couldn't have a circle with a circumference of exactly one meter. Of course, to do so it would have to have a radius of irrational length, but you can't have everything.

First of all, Pi appears to be normal (that is the digits actually meet certain statistical tests for randomness). That is a pattern in some sense. In any event, digits to any base (even base 2 or base 3) are in many ways a very artificial way of thinking about numbers. A far more natural way is to represent numbers as continued fractions http://en.wikipedia.org/wiki/Continued_fraction [wikipedia.org], When considering generalized continued fractions, Pi has a variety of different very elegant patterns.

heu? what is the point of dividing by 113 ? being able to revert back to radiant "easily".
I would thought we use 360 because it is divisible by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360 which is fairly convenient.

The pattern just isn't in base 10. It's in base e. Why does anyone expect to see a numerical pattern in an arbitrary number base like 10? Just because we have 10 fingers doesn't make it the "correct" base for anything.

There's a good argument that the choice of pi = (circumference / diameter) was unfortunate; it should have been (circumference / radius). In the light of modern mathematics it seems clear that the radius is more "fundamental" than the diameter; choosing pi = (circumference / radius) = 6.28... gives a number of nice things like:
A = (1/2)pi r^2, just as E = (1/2)m v^2 or d = (1/2)a t^2, and for the same reason.
In general, in the current convention, 2pi seems to show up a lot more than pi, e.g. there are 2pi radians in a circle, sin(x) has period 2pi, etc. All these would become simply pi with the (circumference / radius) convention

Yeah. Pi acts like Infinite Monkeys. All _we_ have to do is to point to the monkey that actually does write Shakespeare, i.e.: the index of Pi which actually represents Kill Bill Complete in AVI format.

The only problem is the size of that index, but hey, if you zip that number and take its MD5, you have achieved something similar to this [wikipedia.org].

It's a great way to test the performance of these supercomputers, to ensure that their calculations are correct. The calculation of pi to additional decimal places beyond what was previously known is never done with just a single method--otherwise, it is impossible to verify the additional digits. It is always done with two different algorithms to ensure that the result is valid. There are many rapidly converging algorithms (e.g., variations on AM-GM methods can be quadratically convergent or better; BBP-type digit extraction methods; and of course, classic Ramanujan series-type methods). However, computing pi to so many decimal places has much less to do with the chosen algorithm than it has to do with the memory- and computing time-efficient implementations of such algorithms in massively parallel architectures. Thus these calculations serve as very good tests for the robustness of supercomputers. The result is also verifiable to previously known digits, and even beyond the previous record, it is possible to perform statistical analyses to determine whether there are any significant deviations in the distribution of digit frequencies.

So, in summary, it is hardly a useless computation. Not that you're going to get an explanation like this from your usual news sources, which generally do not write for technical audiences.

Also note that distributed computing resources such as Folding@home, or even the Great Internet Mersenne Prime Search don't bother with calculating pi, as the purpose of these projects is to make new discovers in their respective fields of interest.

Well, I'm not a mathematician, but it seems to me that's precisely why there isn't a repetitive pattern in the numerical representation. If there was, that would mean the ratio can be exactly defined by a finite amount of information. It seems to me that asking for a finite decimal represensation of pi is similar to asking someone to exactly represent a circle out of line segments (or to exactly define a circle using a finite set of points). The circumference of the circle is the sum of the length of line segments delineating the circle. The problem is that you need infinitely many of them to exactly define the circle.

Well, I'm not a mathematician, but it seems to me that's precisely why there isn't a repetitive pattern in the numerical representation. If there was, that would mean the ratio can be exactly defined by a finite amount of information.

It can be exactly defined by a finite amount of information. And it's not impossible, in general, for a transcendental number to have some sort of pattern in the numerical representation. For instance, the Champernowne constant --.12345678910111213...

Actually, the program itself is a perfectly fine way of representing pi. See: computable numbers [wikipedia.org]. Note that almost all [wikipedia.org] real numbers are not computable, so it is a non-trivial property.

I always found the Basel problem to be the most elegant converging series involving pi (being the square root of six times the sum of the reciprocals of the squares), probably because there are so many (elegant) proofs of this [ex.ac.uk] (pdf), because it's so simple to understand yet not so simple to prove on a cursory inspection, and because it's the specific case that generalized to one of the most important unsolved problems in mathematics [wikipedia.org].

Better ways to represent that....
\[4\cdot\sum_{n=0}^{\infty}\left(\frac{\left(-1\right)^{n}}{2\cdot n+1}\right)=\pi\]
was trying for a more elegant representation, but I'm going to first have to figure out how to make slashdot accept mathml...

Ever stopped to think that throwing more computing power at a problem is about as productive as throwing more money at a problem or more man power? You can only do so much before an effort becomes either redundant or the return on investment is as dismal as the stock market has been this past year.

I don't honestly know what the practical value of knowing Pi to the 2.5 trillionth digit is but I'd like to think that there are enough resources in play that the fight f

Egads, I'm sorry to dump on you but I remember when posters on slashdot knew their calculus 101 and some really elementary facts about numbers. If pi had a repeating pattern, it would be a rational number. If it was a rational number, that pattern would appear in any number base, it's a simple property of numbers that has nothing to do with the base you express it in.